Former options guy myself and was thinking about doing some similar writeups on options pricing.
i like the graphs of you drew of the strips with different times to expiraiton. One thing that is generally counterintuitive for people is how there is translation of the graph on the x axis over time (theta and carry rates).
also generally, I think introducing black scholes is good, but doesn't really "stick" for most people for a while. it takes a while to develop an intuition about the components of "C = SN(d1) - KN(d2)" means.
I think teaching these intuitions would really sit well for the HN crowd. Also explaining option pricing as the value of the cash flows of the hedge portfolio with constant or regular hedging (like physics, start with assumptions like GBM, zero transaction cost, no bid-ask spread -- then if you ever feel like it later, address how hedge strategies can dramatically change option pricing). Natenberg does a simplied version of this.
Also, Cox Ross Rubenstein is a great way to teach euro AND american options -- just change the rules at each node to get different rules and you can demonstate convergence to Black Scholes with the right parameters.
Do you think there is a market for teaching this? I've taught a couple of courses on this stuff "professionally" and have been debating for a while if I should create some nice online course on the topic.
I suspect there is some interest, but I don't know is the market is big enough to justify creating the materials solely for profit.
As other comments have noted, there are already exhaustive guides on the subject and they are very good. I think the key would be to have a very different approach to it specifically, teach intuitions, not technical details.
hi daleroberts, i'd love to chat with you about this. since i responded to your post i decided i am interested in crafting a short ebook in this vein and would love to chat with you. ezl@rocketlease.com, if you're open to discussing.
Imagine a drunk man walking down a dark, rainy road one night after his car breaks down. He gets out, and begins a "random" walk along the road, looking for help. Neither you (the trader), nor he (the stock), can see anything - it's raining, and it's dark. The man stumbles along the road and your job is to predict his future path, and in return, you get paid money.
A lot of money.
Here's the problem: the guy is both drunk and blind (and probably just a little stupid) - he can't see anything and hence his movements are erratic. How in the hell can you predict where this idiot is going to end up? You have bills to pay, derivatives to price and insurance to sell. So you latch onto the closest thing that'll work - volatility (anchoring bias).
You use his volatility and assume that, dependant upon his past movements, and in turn his apparent level of drunkenness, you can, more or less, predict the possible range of his future random walk. This is a very useful model for predicting where he will be within the next 30 seconds. It works very well, and you make a lot of money.
It feels good.
Now you become confident. You start projecting it out just that little bit further, putting on more precise predictions with tighter spreads, and levering up your bets - because, of course, everyone else is competing with you and driving down your alpha. You have now mistaken past movement for the actual risk of movement - they are not the same thing.
Unfortunately for you, of course, the man has broken down on the edge of a steep cliff. He continues his random walk, blissfully unaware of his impending doom. You continue your bets on his volatility. I mean, why wouldn't you? You're king of the fucking world after all - in fact, not only are you rich, but you also have a Nobel Prize in Economics from a Swedish Bank!
Oops - too late. Your man has just fallen off the cliff, and you, and your savings, along with him. You yell inefficient markets, beta sucks balls, VAR is a trap, the CAPM is a lie and modern portfolio theory is fucking stupid. The last thing we hear, before that final, brutal, resounding thud is the faint line: "It was all a fucking lie."
You have just met real risk. It has not been a pleasant experience. Welcome to the real world.
This is just plain rhetoric and has nothing to do with real world trading. Anybody trading options who takes an outright view is doing it wrong (unless they have extremely good reason to do so).
The market is not dumb and prices volatility accordingly (hence a skew).
What you're talking about is a tail event - most sane option strategies protect against tail events (unless you're writing options and not covering your behind).
Most views via options are on spreads or on volatility where the maximum downside is known (in fact, betting in favor of a tail event is an extremely cheap strategy - via a strangle centered around the current spot - the deeply-out-of money options trade near zero while if a tail event does happen, you're entitled to a handsome payout - the max you lose is the amount you paid upfront - which might be as little as a few hundred dollars - essentially it's a lottery).
So even though your underlying stock may fall off a cliff, you wouldn't care if you had a sane option portfolio because what you get paid for is the distance covered by the drunkard - whether he covers it by falling off a cliff or going zig-zag ad infinitum doesn't matter.
Professional options traders prey on ineffiecent markets, never curse them. Beta and CAPM literally have nothing to do with options trading, pricing, or theory. VAR, like all models, is a trap. Options traders would know this; they are ingrained with a deep distrust of models since they forced to use them all day every day and understand their appropriate uses and limitations. It would be unreasonable to expect any serious options trader to blame her model or worse, "all of it", for failure.
Suggestion:
In practice, for stock-options, volatility is strike-dependent ie implied volatility has a skew (lower strikes have higher implied volatility - the leverage effect). Also, the volatility is dependent on time to expiry of the option.
Also, as I've mentioned in the HN thread for your previous article, single-name stock options which are exchange-traded
have an American exercise type. The valuation of these cannot be done using Black-Scholes if they are (a) long-dated (ie time to expiry is quite long) or (b) they are deeply in the money/out of money or (c) have an underlying which has a significant dividend yield (in which case you'd want to own the stock rather than the option).
Puts behave different than calls if the exercise-style is American (puts have a limited upside - so you wouldn't wait too long if the underlying stock has fallen far enough - your payoff is not likely to be larger).
You may want to discuss this and option greeks the next time.
The first thirty or so pages offer the most intuitive introduction to vanillas I've read. The middle is must read, and the end gets into exotics if that's your fancy.
The math looks off on the 50% vol 30 day option. No way a $50-strike option on a stock at $25 would be pricing at $20. Unless you mean 50% non-annualized vol, which would not be very conventional.
I really like this series of posts--it's a great introduction to options.
The little aside in one of the notes made me curious: what is the logic behind trading options on their expiry date? I can't think of any obvious reason to do that, but I'm not terribly experienced with finance. That's why I'm curious about it, I suppose.
a lot of people DON'T trade options on their expiration day. Even marketmakers, whose job is to provide liquidity in the name often shy away from it or demand high premia.
historically, retail investors sometimes closed out their positions on expiration day instead of exercising. this might sound weird, but some people have weird portfolio restrictions so its occasionally sensible.
also, for some equities, there are big events that happen to coincide with expirations. If you have an opinion on a big event (lets say AAPL options are expiring and you think there's going to be a huge announcement), it lets you buy expiring options for virtually no volatility premium.
also: people like gambling. buying cheapie out-of-the-money options is kind of like buying a lottery ticket. I've seen them hit (saw a guy take home a 5million dollar win one wednesday morning for a $6k lottery ticket he bought on the close on tuesday)
I think the author of the article hasn't really grasped the concept of option pricing. You are not calculating an 'expected value'.
The whole point of the theory is there is a correspondence between a 'no-arbitrage argument' and calculating an expectation under the 'risk-free' measure. The mathematical operation of expectation E is only a tool. This link between no-arbitrage and martingale theory is called the 'Fundamental Theorem of Asset Pricing'.
There is a nice (basic) explanation of this idea in the book by Baxter and Rennie "Financial Calculus" where they compare the 'expected value' approach of a bookmaker and the 'no-arbitrage' approach.
For a more advanced explanation, you can have a look at the book by Delbaen and Schachermayer (2006).
Of course you're calculating an expected value. If the expected value of an option isn't its price, what is it?
The no-arbitrage argument is another way of looking at it, but the two methods are equivalent. In particular, if the expected value of an option is higher than its price, you should buy the option - and if it's lower, you should sell it.
With just one transaction this would be statistical arbitrage rather than pure arbitrage, but if option prices regularly differed from the option's expected value, stat arb would be a fine strategy.
Sorry if my comment came across harshly, I didn't intend it to.
By expected value, I mean the price as you'd calculate it with a risk-neutral valuation based on some model of the underlying security.
For example, if you have a model that says an option is worth $4, and it's selling for $2, you should buy it if you're confident in your model. If you can do this repeatedly on a bunch of independent options you'll make money in the long run (assuming your model is correct and you're placing a large enough number of bets relative to the probability of making money on an individual option).
I learned this with a combination of practical experience, self-study, and coursework.
I feel like this could have been longer. Volatility is really the only way options make money, no matter what your position is. Is volatility going to be covered in later parts as well?
I think a discussion of measuring and predicting volatility, and how volatility varies based on the class of asset would be useful. Someone who invests but doesn't know a lot about options probably doesn't have the respect for volatility that an options trader does.
I'm going to get flamed for this, but promulgating Black-Scholes as the "standard way" to price options is a little misleading. A better choice of word would be "conventional", but that's also inexact because the BSM is broken, since BSM makes a lot of unrealistic assumptions for the model to work. The point is that there is no One True Pricing Model... the price is what the market thinks the value is. Too many people who should know better get it all mixed up, with disastrous effects.
It's not really flame-worthy because you're mostly right. I do however, think Black-Scholes is an appropriate starting point for options theory, even if it's not used a ton in practice. Without some sort of framework, you're effectively lost. I doubt many people could realistically develop any sort of options based strategy (besides simple leveraged long / short of the underlying) without a deep inside-and-out understanding of the BSE. The reason being not that you need to use it to calculate anything, but because it lays out a straightforward conceptual model of how various greeks interact with each other.
Of course, understanding the shortfalls of it are equally important. But the same could be said of DCF, P/E or any other valuation method.
Former options guy myself and was thinking about doing some similar writeups on options pricing.
i like the graphs of you drew of the strips with different times to expiraiton. One thing that is generally counterintuitive for people is how there is translation of the graph on the x axis over time (theta and carry rates).
also generally, I think introducing black scholes is good, but doesn't really "stick" for most people for a while. it takes a while to develop an intuition about the components of "C = SN(d1) - KN(d2)" means.
I think teaching these intuitions would really sit well for the HN crowd. Also explaining option pricing as the value of the cash flows of the hedge portfolio with constant or regular hedging (like physics, start with assumptions like GBM, zero transaction cost, no bid-ask spread -- then if you ever feel like it later, address how hedge strategies can dramatically change option pricing). Natenberg does a simplied version of this.
Also, Cox Ross Rubenstein is a great way to teach euro AND american options -- just change the rules at each node to get different rules and you can demonstate convergence to Black Scholes with the right parameters.